He continues to be dogged by accusations of ducking scrutiny after failing to take part in a Sky News debate this week.

Mr Hunt told Sky News: “This is an audition to be prime minister of the UK. If Boris is refusing to answer questions in the media, refusing to do live debates then of course people are thinking: just who are we going to get as PM?”

Jeremy Hunt has told Mr Johnson: "Don't be a coward" (Picture: PA)

Tory MP Tobias Ellwood warned a Johnson-led Government could be brought down within days.

Mr Ellwood said ‘a dozen or so’ Conservative MPs would consider bringing down a Tory government rather than see the UK leave the EU without a deal.

If a number of rebel Tories supported a no-confidence motion the government would collapse and a General Election would be triggered.

Mr Johnson says he would take the UK out of the EU without a deal on 31 October if he fails to negotiate an improved deal before then.

The Conservative government currently has a majority of just four, meaning only a very small number of MPs need to rebel to tear the government down.

Media outside the home of Conservative party leadership candidate Boris Johnson in south London (Picture: PA)

The former foreign secretary’s personal life is being dragged into the race to be the next Prime Minister after the police incident at the weekend.

The Daily Mirror reported that, according to friends, Mr Johnson wants to get back with his estranged wife, barrister Marina Wheeler, despite being in a relationship with Ms Symonds, and that he is finding their divorce “extremely painful”.

Sources close to his family told the newspaper that he was “all over the place” and “psychologically unfit” to be Ms Symonds’s long-term partner.

Meanwhile, The Sun reported that Mr Johnson and Ms Symonds have had four serious rows in the past six weeks and nearly split up at the start of this month.

A friend told the newspaper: “Carrie and Boris have an incredibly volatile relationship — this was by no means the first explosive row they’ve had.”

Labour MP Jess Phillips said Mr Johnson should have taken the opportunity at a leadership hustings event on Saturday to thank his neighbours for being concerned for the welfare of his partner.

She told LBC: "Boris Johnson had the opportunity to give an explanation and to say to the nation that it was the right thing for the neighbours to call the police and the right thing for them to try to gather evidence.

"But instead his very poor character has picked himself over the safety of women in this country, and the line that he is putting out is that this is a private family matter.

"Those of us who have campaigned for years have tried to move that dial, and he is taking us back and that is dangerous."

Mr Hunt urged Mr Johnson not to be “a coward” about facing public scrutiny.

Writing in The Times, he insisted he has no interest in debating Mr Johnson’s private life, but wants to challenge him on television over his commitment to taking the UK out of the EU by the end of October.

Boris Johnson and Jeremy Hunt sit side by side during last week's TV debate (Picture: Getty)

Mr Hunt said: "A new prime minister needs the legitimacy of having made his arguments publicly and having them subjected to scrutiny.

"Only then can you walk through the front door of No 10 with your head held high instead of slinking through the back door, which is what Boris appears to want."

He added: "Don't be a coward Boris, man up and show the nation you can cope with the intense scrutiny the most difficult job in the country will involve."

Mr Hunt said on Sunday he thought any candidate for prime minister "should answer questions on everything".

Boris Johnson and Jeremy Hunt are fighting to be the next prime minister (Picture: Getty)

In his weekly column in The Telegraph, Mr Johnson did not address the incident. Instead, he repeated his determination to deliver Brexit by Halloween.

He wrote: "We must leave the EU on Oct 31 come what may. It will honour the referendum result, it will focus the minds of EU negotiators."